A Superlinear Lower Bound on the Number of 5-Holes

Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P . For a positive integer n, let h5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h5(n) have been of order Ω(n) and O(n2), respectively. We show that h5(n) = Ω(n log4/5 n), obtaining the first superlinear lower bound on h5(n). ∗ The research for this article was partially carried out in the course of the bilateral research project “Erdős–Szekeres type questions for point sets” between Graz and Prague, supported by the OEAD project CZ 18/2015 and project no. 7AMB15A T023 of the Ministry of Education of the Czech Republic. Aichholzer, Scheucher, and Vogtenhuber were partially supported by the ESF EUROCORES programme EuroGIGA – CRP ComPoSe, Austrian Science Fund (FWF): I648-N18. Parada was supported by the Austrian Science Fund (FWF): W1230. Balko and Valtr were partially supported by the grant GAUK 690214. Balko, Kynčl, and Valtr were partially supported by the project CE-ITI no. P202/12/G061 of the Czech Science Foundation (GAČR). Hackl and Scheucher were partially supported by the Austrian Science Fund (FWF): P23629-N18. Balko, Scheucher, and Valtr were partially supported by the ERC Advanced Research Grant no. 267165 (DISCONV). © Oswin Aichholzer, Martin Balko, Thomas Hackl, Jan Kynčl, Irene Parada, Manfred Scheucher, Pavel Valtr, and Birgit Vogtenhuber; licensed under Creative Commons License CC-BY 33rd International Symposium on Computational Geometry (SoCG 2017). Editors: Boris Aronov and Matthew J. Katz; Article No. 8; pp. 8:1–8:16 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 8:2 A Superlinear Lower Bound on the Number of 5-Holes The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line ` into two subsets, each of size at least 5 and not in convex position, then ` intersects the convex hull of some 5-hole in P . The proof of this result is computer-assisted. 1998 ACM Subject Classification G.2.1 Combinatorics